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4.2.1 Linear Shape Functions on Tetrahedral Elements

Figure 4.7: Tetrahedral element.
\includegraphics[width=14cm]{figures/fem/tetrahedron.eps}

The unknown field in the tetrahedral element (Fig. <4.7>) is approximated by the linear function

$\displaystyle \phi(x,y,z) = a + bx + cy + dz.$ (4.58)

Analogously to the two-dimensional case the four coefficients $ a$ , $ b$ , $ c$ and $ d$ are obtained assuming that the field values $ \phi_1$ , $ \phi_2$ , $ \phi_3$ and $ \phi_4$ on the four vertexes of the tetrahedron are known.

\begin{displaymath}\begin{split}\phi_1& = a + bx_1 + cy_1 + dz_1  \phi_2& = a ...
... + cy_3 + dz_3  \phi_4& = a + bx_4 + cy_4 + dz_4. \end{split}\end{displaymath} (4.59)

The coordinates $ x_i$ , $ y_i$ and $ z_i$ correspond to the vertex $ i$ . From (4.59) the coefficients $ a$ , $ b$ , $ c$ and $ d$ become

\begin{displaymath}\begin{split}a = \frac{1}{J} \left\vert\begin{array}{cccc} \p...
...uad  a_1\phi_1 + a_2\phi_2 + a_3\phi_3 + a_4\phi_4 \end{split}\end{displaymath} (4.60)

\begin{displaymath}\begin{split}b = \frac{1}{J} \left\vert\begin{array}{cccc} 1 ...
...uad  b_1\phi_1 + b_2\phi_2 + b_3\phi_3 + b_4\phi_4 \end{split}\end{displaymath} (4.61)

\begin{displaymath}\begin{split}c = \frac{1}{J}\left\vert\begin{array}{cccc} 1 &...
...uad  c_1\phi_1 + c_2\phi_2 + c_3\phi_3 + c_4\phi_4 \end{split}\end{displaymath} (4.62)

\begin{displaymath}\begin{split}d = \frac{1}{J}\left\vert\begin{array}{cccc} 1 &...
...ad  d_1\phi_1 + d_2\phi_2 + d_3\phi_3 + d_4\phi_4. \end{split}\end{displaymath} (4.63)

The Jacoby determinant for the three-dimensional case is given by

$\displaystyle J = \left\vert \begin{array}{cccc} 1 & x_1 & y_1 & z_1  1 & x_2...
... z_4 \end{array} \right\vert = (\vec{r}_1\times\vec{r}_2)\cdot\vec{r}_3 = 6V_e,$ (4.64)

where $ V_e$ is the volume of the tetrahedron. Equations (4.60) to (4.63) define the auxiliary coefficients $ a_i$ , $ b_i$ , $ c_i$ and $ d_i$ ($ i\in[1;4]$ ), which are used to write (4.58) as

\begin{displaymath}\begin{split}\phi(x,y,z) & = (a_1 + b_1x + c_1y + d_1z)\phi_1...
..._4y + d_4z)\phi_4 = \sum^4_{i=1}\lambda^e_i \phi_i \end{split}\end{displaymath} (4.65)

to introduce the element shape functions $ \lambda^e_i$ .


next up previous contents
Next: 4.2.2 Tetrahedron Barycentric Coordinates Up: 4.2 Three-Dimensional Scalar Finite Previous: 4.2 Three-Dimensional Scalar Finite   Contents

A. Nentchev: Numerical Analysis and Simulation in Microelectronics by Vector Finite Elements